The analogy between electromagnetic waves and linear elastodynamics

Physical analogies aid students to transfer knowledgment from familiar phenomena to more abstract physical concepts, specially when the mathematics is similar. Here we present an interesting analogy between linear elastic fields and its governing equations and the more abstract electromagnetic fields. For this, we define the elastic $\mathbf{E}$ and $\mathbf{B}$ fields, and show that the Navier-Cauchy elastodynamic equation can be put in the form of four Maxwell-like equations. We also introduce a elastic potential vector $\mathbf{A}$ and a elastic scalar potential $\varphi$. Of course, the analogy only works in the rest reference frame of the elastic medium, but it seems pedagogical because students easily imagine elastic waves (displacement waves $\mathbf{u}(\mathbf{r},t)$ in an elastic material) but have dificulty to visualize electromagnetic fields. We also show that, although the elastic vector potential $\mathbf{A}$ is not gauge invariant, the displacement $\mathbf{u}=\mathbf{A}\nabla \kappa +\mathbf{A}$ is gauge invariant (as a measurable physical quantity should be), where the scalar field $\kappa$ is related to $\varphi$ by $\varphi =\frac{1}{c}{\partial }_t\kappa$.

Keywords:
Electromagnetism; Linear Elasticity; Maxwell’s Equations; Elastodynamics; Navier-Cauchy Equations; Gauge Invariance